Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

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April 2012, 6(2): 205-249. doi: 10.3934/jmd.2012.6.205

Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

Alexandra Monzner ^1, , Nicolas Vichery ^2, and Frol Zapolsky ^3,

1.	Fakultät für Mathematik, TU Dortmund, Dortmund, Germany
2.	CMLS École Polytechnique, Palaiseau, France
3.	Mathematisches Institut der Ludwig-Maximilian-Universität, Munich, Germany

Received January 2012 Published August 2012

Abstract
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For a closed connected manifold $N$, we construct a family of functions on the Hamiltonian group $\mathcal{G}$ of the cotangent bundle $T^*N$, and a family of functions on the space of smooth functions with compact support on $T^*N$. These satisfy properties analogous to those of partial quasimorphisms and quasistates of Entov and Polterovich. The families are parametrized by the first real cohomology of $N$. In the case $N=\mathbb{T}^n$ the family of functions on $\mathcal{G}$ coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of $\mathcal{G}$, to Aubry--Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.

Mathematics Subject Classification: Primary: 53D40, 37J5.

Citation: Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205

References:

[1]	A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722. doi: 10.2140/gt.2010.14.1569.
[2]	P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology,, Int. Math. Res. Not. IMRN, 2008 ().
[3]	A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. doi: 10.1007/BF02566074.
[4]	P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.
[5]	M. Brunella, On a theorem of Sikorav, Ens. Math. (2), 37 (1991), 83-87.
[6]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology,, Int. Math. Res. Not., 2003 (): 1635. doi: 10.1155/S1073792803210011.
[7]	M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[8]	M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.
[9]	M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), part 1, 1037-1055.
[10]	U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math., 159 (2007), 1-56. doi: 10.1007/s11856-007-0037-3.
[11]	R. Iturriaga and H. Sánchez-Morgado, A minimax selector for a class of Hamiltonians on cotangent bundles, Internat. J. Math., 11 (2000), 1147-1162.
[12]	S. Lanzat, "Symplectic Quasi-Morphisms and Quasi-States for Noncompact Symplectic Manifolds,", Ph. D. thesis, ().
[13]	S. Lanzat, Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds,, , ().
[14]	R. Leclercq, Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn., 2 (2008), 249-286. doi: 10.3934/jmd.2008.2.249.
[15]	J. N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems, Math. Z., 207 (1991), 169-207 doi: 10.1007/BF02571383.
[16]	D. Milinković and Y.-G. Oh, Floer homology as the stable Morse homology, J. Korean Math. Soc., 34 (1997), 1065-1087.
[17]	D. Milinković and Y.-G. Oh, Generating functions versus action functional. Stable Morse theory versus Floer theory, in "Geometry, Topology, and Dynamics" (Montreal, PQ, 1995), CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, (1998), 107-125.
[18]	A. Monzner and F. Zapolsky, A comparison of symplectic homogenization and Calabi quasi-states, J. Topol. Anal., 3 (2011), 243-263. doi: 10.1142/S1793525311000581.
[19]	Y.-G. Oh, Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Diff. Geom., 46 (1997), 499-577.
[20]	Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants, Comm. Anal. Geom., 7 (1999), 1-54.
[21]	Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in "The Breadth of Symplectic and Poisson Geometry," Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 525-570.
[22]	G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Mosc. Math. J., 3 (2003), 593-619, 745.
[23]	L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphisms," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.
[24]	L. Polterovich and K. F. Siburg, On the asymptotic geometry of area-preserving maps, Math. Res. Lett., 7 (2000), 233-243.
[25]	P. Py, Quelques plats pour la métrique de Hofer, J. Reine Angew. Math., 620 (2008), 185-193. doi: 10.1515/CRELLE.2008.053.
[26]	M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.
[27]	K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 92 (1998), 295-319. doi: 10.1215/S0012-7094-98-09207-9.
[28]	K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics," Lecture Notes in Mathematics, 1844, Springer-Verlag, Berlin, 2004. doi: 10.1007/b97327.
[29]	A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403. doi: 10.2140/gt.2010.14.2383.
[30]	D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology Appl., 96 (1999), 249-266. doi: 10.1016/S0166-8641(98)00049-2.
[31]	Viterbo, C., Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685-710. doi: 10.1007/BF01444643.
[32]	C. Viterbo, Symplectic homogenization,, , ().
[33]	F. Zapolsky, On the Lagrangian Hofer geometry in symplectically aspherical manifolds,, , ().

show all references

References:

[1]	A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722. doi: 10.2140/gt.2010.14.1569.
[2]	P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology,, Int. Math. Res. Not. IMRN, 2008 ().
[3]	A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. doi: 10.1007/BF02566074.
[4]	P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.
[5]	M. Brunella, On a theorem of Sikorav, Ens. Math. (2), 37 (1991), 83-87.
[6]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology,, Int. Math. Res. Not., 2003 (): 1635. doi: 10.1155/S1073792803210011.
[7]	M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[8]	M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.
[9]	M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), part 1, 1037-1055.
[10]	U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math., 159 (2007), 1-56. doi: 10.1007/s11856-007-0037-3.
[11]	R. Iturriaga and H. Sánchez-Morgado, A minimax selector for a class of Hamiltonians on cotangent bundles, Internat. J. Math., 11 (2000), 1147-1162.
[12]	S. Lanzat, "Symplectic Quasi-Morphisms and Quasi-States for Noncompact Symplectic Manifolds,", Ph. D. thesis, ().
[13]	S. Lanzat, Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds,, , ().
[14]	R. Leclercq, Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn., 2 (2008), 249-286. doi: 10.3934/jmd.2008.2.249.
[15]	J. N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems, Math. Z., 207 (1991), 169-207 doi: 10.1007/BF02571383.
[16]	D. Milinković and Y.-G. Oh, Floer homology as the stable Morse homology, J. Korean Math. Soc., 34 (1997), 1065-1087.
[17]	D. Milinković and Y.-G. Oh, Generating functions versus action functional. Stable Morse theory versus Floer theory, in "Geometry, Topology, and Dynamics" (Montreal, PQ, 1995), CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, (1998), 107-125.
[18]	A. Monzner and F. Zapolsky, A comparison of symplectic homogenization and Calabi quasi-states, J. Topol. Anal., 3 (2011), 243-263. doi: 10.1142/S1793525311000581.
[19]	Y.-G. Oh, Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Diff. Geom., 46 (1997), 499-577.
[20]	Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants, Comm. Anal. Geom., 7 (1999), 1-54.
[21]	Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in "The Breadth of Symplectic and Poisson Geometry," Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 525-570.
[22]	G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Mosc. Math. J., 3 (2003), 593-619, 745.
[23]	L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphisms," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.
[24]	L. Polterovich and K. F. Siburg, On the asymptotic geometry of area-preserving maps, Math. Res. Lett., 7 (2000), 233-243.
[25]	P. Py, Quelques plats pour la métrique de Hofer, J. Reine Angew. Math., 620 (2008), 185-193. doi: 10.1515/CRELLE.2008.053.
[26]	M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.
[27]	K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 92 (1998), 295-319. doi: 10.1215/S0012-7094-98-09207-9.
[28]	K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics," Lecture Notes in Mathematics, 1844, Springer-Verlag, Berlin, 2004. doi: 10.1007/b97327.
[29]	A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403. doi: 10.2140/gt.2010.14.2383.
[30]	D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology Appl., 96 (1999), 249-266. doi: 10.1016/S0166-8641(98)00049-2.
[31]	Viterbo, C., Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685-710. doi: 10.1007/BF01444643.
[32]	C. Viterbo, Symplectic homogenization,, , ().
[33]	F. Zapolsky, On the Lagrangian Hofer geometry in symplectically aspherical manifolds,, , ().

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